Refresher - Commutative, Associative, Distributive Properties
Closure Property: A set is closed (under an hoperation) if and only if the operation on two elements of the set produces another element of the set. If an element outside the set is produced, then the operation is not closed. The set of even numbers fm a closed set under addition and multiplication, since when you add two evens, or multiply two evens, you get an even number. The odd numbers are closed under multiplication but not addition. If you add two odd numbers together, say 5 and 7, the result, 12 is not odd.
Commutative Property: When an operation has this property, it means that the order of elements does not change the outcome of the operation. Subtraction is not a commutative operation because a-b is not te same as b-a. However, addition and multiplication are both commutative operations because a+b=b+a and ab=ba are true for every a and b in the real numbers.
In some everyday situations, we can find instances where order does not affect the final outcome. For instance, if we wash our hands and then wash our face, the outcome will not be different if the order is reversed so that we wash our face first, then our hands. However, there are also instances where order does affect the outcome: putting on socks and then shoes will have a different result than putting on shoes then socks.
Associative Property: "Associative" comes from "associate" or "group", and the Associative Property is the one that refers to grouping without changing order. For addition, the property is illustrated by a+(b+c)=(a+b)+c while in multiplication we have a(bc)=(ab)c for every a, b and c in the real numbers. A concrete example would be 2+(3+4)=(2+3)+4 or 2 x (3x4)=(2x3) x 4. Note: Subtraction for the set of integers is not associative since (2 -3) - 4 = -5 where as 2 - (3-4) = 3
Distributive Property: The Distributive Property describes the interaction of the two operations of multiplication and addition. So a(b+c)=ab+ac. We say multiplication distributes over addition. Because the operation of subtraction is defined through addition, by the process of adding the opposite, it is a natural extension to see that multiplication also distributes over subtraction. The Distributive Property can be used to simplify calculations and to do mental arithmetic, as these examples show:
17(101)=17(100+1)=17(100) + 17(1)=1717
23(998)=23(1000-2) =23(1000)-23(2)=2254
Take the operation of the Greatest Integer together with addition. We can test whether the Distributive Property holds in this case. [(a+b)]=?[a]+[b]. If we can find one counter example, then we have shown it does not hold in general. Take a=4.5 and b=5.5. Then [(4.5 + 5.5)] = [10] = 10, but [4.5] + [5.5] = 4 + 5 = 9, so this is a case where the distributive property doesn't work.
Multiplication is also distributive over subtraction. This can help us with mental arithmetic. Say you wanted to multiply: 35 x 99. Just re-write: 35 (100 – 1) = 3500 – 35 = 3465.
Note: the distributive property does not automatically work over all operations. Is 8 * (4 / 2) equal to (8 * 4) / (8 * 2)? Of course not!
Identity Property: The Identity Property refers to the existence of a special element in a set that goes along with a particular operation. If we perform the operation between the special element and any other in the set, then the second element retains its original value, or identity. For addition, 0 is the identity element, and for multiplication 1 is the identity element. This is because the addition of 0 to any number does not change it, and the multiplication of 1 to any number does not change it, either.
The determination of the pH of a substance is the measurement of the H+ ions found in that particular substance. pH is determined and recorded as a number between 0 and 14. Pure de-ionized water has a pH of 7 which is neutral and acts like an identity element, because the addition of water to any other element has no effect on the resulting pH of the mixture
Inverse Property: The Inverse Property works hand in hand with the Identity property.
A non-zero element, X, of a set has an inverse with respect to a given operation if and only if there exists another member, Y, of the same set, such that when the operation is performed the identity element is produced. For the set of real numbers 2 is the inverse of -2 for the operation of addition and -½ is its inverse for the operation of multiplication. We see that every real number except 0 has both of these kinds of inverses, which we call opposites, and reciprocals, respectively.
In chemistry, if the level of OH- ions increases, the substance is considered to be alkaline or base and the pH number is above 7. An acid has a range of 0 to any numerical value below 7. For example, 6.9 would be a weak acid. A base has a range of any numerical value above 7 to 14 with 7 being a neutral value. Inverse elements exists when we mix a certain base with a certain acid to produce a pH of 7.
In our road map example, traveling two miles to the south would be the inverse of traveling two miles to the north. Combining the two would leave our traveler in the same space.
If you need to review, select any of the topics in the navigation box. When you're ready, on the next screen, we'll begin to put these building blocks together as we look at field theory.